Differential phase extraction in dual interferometers exploiting the correlation between classical and quantum sensors

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Differential phase extraction in dual interferometers exploiting the correlation between classical and quantum

sensors

Mehdi Langlois, Romain Caldani, Azer Trimeche, Sébastien Merlet, Franck Pereira dos Santos

To cite this version:

Mehdi Langlois, Romain Caldani, Azer Trimeche, Sébastien Merlet, Franck Pereira dos Santos. Dif-

ferential phase extraction in dual interferometers exploiting the correlation between classical and

quantum sensors. Physical Review A, American Physical Society, 2017, 96 (5), �10.1103/Phys-

RevA.96.053624�. �hal-01655416�

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between classical and quantum sensors

M. Langlois, R. Caldani, A. Trimeche, S. Merlet, and F. Pereira dos Santos

^∗

LNE-SYRTE, Observatoire de Paris, PSL Research University,

CNRS, Sorbonne Universit´ es, UPMC Univ. Paris 06, 61 Avenue de l’Observatoire, 75014 Paris, France

We perform the experimental demonstration of the method proposed in [Phys. Rev. A 91, 063615 (2015)] to extract the differential phase in dual atom interferometers. From a single magneto-optical trap, we generate two atomic sources, vertically separated and free-falling synchronously, with the help of an accelerated lattice. We drive simultaneous Raman interferometers onto the two sources, and use the correlation with the vibration signal measured by a seismometer to extract the phase of each interferometer. We demonstrate an optimal sensitivity of the extracted differential phase between the two interferometers, free from vibration noise and limited by detection noise, when the two interferometers are in phase.

I. INTRODUCTION

Quantum sensors based on light pulse atom interfer- ometry [1], such as gravimeters and gyrometers, have demonstrated high level of sensitivities and accuracies, comparable or better than conventional sensors [2–5].

They find today applications in various fields, from fun- damental physics to geophysics, and the transfer of this technology to the industry led in the last years to the de- velopment of commercial atomic gravimeters. The sensi- tivity of these sensors is limited in most cases by vibra- tion noise, whose influence can be mitigated using pas- sive isolation techniques [6], or auxiliary classical sensors for active isolation [7–9], noise correction [6, 10] or hy- bridization [11]. Nevertheless, when the measurand is derived from a differential measurement, performed on two interferometers interrogated at the same time, the vibration noise, which is then in common mode, can be efficiently suppressed. This technique has been used for the measurement of gravity gradients [12, 13] and the precise determination of G [14, 15], for the measurement of rotation rates [16–18], for the test of the universality of free fall with cold atoms [19–21]. It is also of interest for the detection of gravitational waves [22–25].

The differential phase, which is the phase difference be- tween the two simultaneous interferometers, can be ex- tracted simply from a fit to the ellipse obtained when plotting parametrically the output signals of the two in- terferometers [26]. This method rejects the vibration noise efficiently but introduces in general large errors in the determination of the differential phase. Meth- ods based on Bayesian statistics, which require an a pri- ori knowledge of the phase noise of the interferometer, have been proven more accurate [27–29]. Other methods, which use a simultaneous third measurement [30], or di- rect extraction of the individual phases [31], also allow for the retrieval of the differential phase with negligible

∗

Electronic address: franck.pereira@obspm.fr

bias.

In this paper, we perform the experimental demon- stration of the alternative method proposed in [32]. The correlation between the individual interferometer mea- surements and the vibration phase estimated from the measurement of an auxiliary seismometer allows us to recover the visibility of the interferometer fringes in the presence of large vibration phase noise and to extract the phase of each interferometer. The differential phase is then simply obtained by subtracting these two phases.

This method of phase extraction, which was first demon- strated in [10] for a single interferometer, has been em- ployed recently in [33] for simultaneous interferometers performed on two different atomic species. In the lat- ter case, and by contrast with the situation we study here, the difference in the scale factors between the two interferometers reduces the correlation between the two extracted phases, degrading the rejection efficiency of the vibration noise. Here, we operate a dual atom interferom- eter on a single atomic species in a gradiometer configu- ration, with two interferometers separated along the ver- tical direction. Having the same scale factors, the better correlation between those two simultaneous interferome- ters enables us to reject more efficiently the common vi- bration noise. We demonstrate an optimal sensitivity in the differential phase extraction, limited by the detection noise, for in-phase operation of the two interferometers.

II. PRINCIPLE OF THE EXPERIMENT

The experimental setup and the time chart of the mea-

surement sequence are displayed in figure 1. We start by

loading with a 2D magneto-optical trap (2D MOT) a 3D

mirror MOT, realized with four independent beams, two

of them being reflected by the surface of a mirror placed

under vacuum. We trap about 3×10

^{8 87}

Rb atoms within

480 ms, and cool them down to about 1.8 µK with a far

detuned molasses before releasing them from the cool-

ing lasers in the |F = 2i hyperfine ground state. Right

after, they are launched upwards using a Bloch elevator

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2 FIG. 1: Scheme of the experimental setup and measurement sequence. Clouds displayed as open circles are in the state

| F = 1i, clouds displayed as full circles in | F = 2i. MW: mi- crowave antenna, SEL: selection, INT: interferometer, DET:

detection.

[34]. They are first loaded in a co-moving lattice, realized with two counterpropagating laser beams, whose inten- sity is adiabatically ramped up to a lattice depth of about 40 E

r

within 200 µs, and whose acceleration follows the Earth gravity acceleration g. The lattice acceleration is then set to about 80g upward by ramping the frequency difference between the two lattice beams up to 4.5 MHz within 2.25 ms. The lattice intensity is then adiabati- cally switched off, leaving the atoms in free fall with an initial velocity of 1.76 m/s. The efficiency of the launch is about 50 %. The launched atoms are then selected in the |F = 1, m

F

= 0i state with a 0.8 ms long microwave pulse followed by a laser pushing pulse which removes the atoms remaining in |F = 2i state. To lift the de- generacy between the different Zeeman sublevels, a bias field of 400 mG is applied. While these atoms are moving upwards, we load a second atomic cloud in the 3D mirror MOT for 100 ms. This second cloud is then cooled down to 1.8 µK and gets released from the molasses beams at the very moment when the first one reaches its apogee.

We then apply the same selection sequence to prepare the second cloud in the |F = 1, m

F

= 0i state. We arranged the second sequence so that the preparation of the second cloud does not affect significantly the first one: the first cloud being in the |F = 1i state is merely perturbed by the scattered light from the MOT lasers (the repumping light in the second MOT is adjusted so as not to be sat- urated), and remarkably, for the launch velocity we use, the second microwave pulse, of 1.8 ms duration, drives a close to 2π pulse on the first atomic cloud. This is due to a favourable variation of the microwave coupling with vertical position. Most of the atoms of the first cloud thus remain in the |F = 1, m

F

= 0i state.

At a delay of 32 ms after the release of the second cloud, we apply a sequence of three counterpropagat- ing Raman pulses, equally separated in time, onto the two free falling atomic clouds. The Raman transitions couple the two hyperfine states |F = 1i and |F = 2i via a two photon transition, and impart a momentum

transfer ~ k

ef f

to the atoms. k

ef f

= k

1

− k

2

is the ef- fective wavevector of the Raman transition, with k

1

and k

2

the wavevectors of the two Raman lasers. The pulse sequence allows to separate, deflect and recombine the atomic wavepackets, creating simultaneously two verti- cally separated Mach Zehnder atom interferometers. The atomic phase-shift at the output of each interferometer is then given by: ∆Φ = φ

1

− 2φ

2

+ φ

3

, where φ

i

is the phase difference between the two counterpropagating Ra- man lasers at the position of the atoms at the i-th Raman pulse. The Raman beams are vertically aligned, which makes these interferometers sensitive to the local grav- ity accelerations [35]. The interferometer phase shifts are given by k

ef f

g

1

T

²

and k

ef f

g

2

T

²

, where g

1

and g

2

are the gravity accelerations at the altitudes of the two clouds, where T is the time separation between consecu- tive Raman pulses. The duration of the π Raman pulse is 8 µs, which corresponds to a Rabi frequency of 62.5 kHz.

The size of the vacuum chamber limits the maximum duration of the interferometers 2T to about 160 ms. Af- ter the interferometer sequence, the two atomic clouds are detected one after the other by fluorescence, using a state selective detection which measures the populations in each of the two output ports of the interferometers, corresponding to the two hyperfine states |F = 1i and

|F = 2i.

The laser system we use for cooling, detecting and driv- ing the interferometer pulses is based on semiconductor laser sources, and is described in detail in [36]. For the Bloch elevator, we generate a lattice laser using frequency doubling techniques (see figure 2). A DFB diode laser at 1560 nm is first amplified by a fiber amplifier up to 5 W, and seeds a high power resonant frequency doubling module (from the company Muquans) which delivers up to 3 W at 780 nm. The frequency of the seed is adjusted such that the frequency doubled light is blue detuned from the

⁸⁷

Rb D2 transition by 50 GHz. The output of the doubler is then split into two beams, each one being frequency shifted with a double pass acousto-optic mod- ulator (AOM) before being recombined with orthogonal linear polarisations using a polarizing beam splitter cube.

The combined beam is diffracted by a last AOM, which is used for switching and controlling the intensity of the lattice beams. The Raman beams, which are derived from the first laser system, have the same linear polari- sation. They are overlapped with the lattice beams using the very same AOM, into which they are injected at an angle, along the direction of the first diffracted order.

This arrangement allows to overlap the diffracted lattice beams with the non-diffracted Raman beams in order to inject them into the same fiber, and takes advantage of the fact that the two beams are not used at the same time. As for the switching of the Raman beams, it is performed with the combination of an AOM and a me- chanical shutter (an optical scanner), both located in the first laser system.

Raman and lattice beams are injected into a common

polarization maintaining fiber, out of which we get a total

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EDFA

PhD

/2

Fiber laser

AOM

/4 ν-δ f νC

/4 ν+δ

f

AOM AOM Raman bench

ν

_R1

+ν

_R2

/2 SHG

FIG. 2: Scheme of the lattice beams laser setup. The lattice beams generated by a frequency doubled fiber telecom laser are combined with the Raman beams using an AOM.

power of 500 mW for the lattice beams (for a doubler output of 2 W), and 26 mW for the Raman beams. The ratio between the intensities of the two Raman beams is adjusted to cancel the differential light shift. The beams get collimated to a 1/e

²

radius of 3.75 mm and enter the vacuum chamber through the top of the experiment. The polarization configuration for these beams is displayed in figure 3.

σ

⁺

σ

⁺

σ

^-

σ

^-

σ

^-

λ/4

LCVR (λ/4)

λ/4

LCVR (0)

σ

^-

σ

^-

FIG. 3: Polarization configuration of lattice (left) and Raman (right) beams in the vacuum chamber. The combination of a quarter-wave plate and an adjustable retarding plate (LCVR) with a polarizing beamsplitter cube allows to obtain the re- quired polarizations for realizing an accelerated lattice along one direction only and for driving counterpropagating Raman transitions.

A fixed quarter wave plate converts the linear polari- sations into circular ones : σ

+

− σ

₋

or σ

₋

− σ

₋

for the lattice or Raman beams, propagating downwards. At the bottom of the chamber, the beams pass through a liq- uid crystal variable retarder plate (LCVR), a polarizing beamsplitter cube (PBC) with proper axes at 45

^◦

with respect to the axes of the LCVR and finally get retrore- flected onto a mirror. The retardance of the LCVR plate

is set to λ/4 during the lattice launch, such that one of the two downward lattice beams gets trashed by the PBC.

Without the cube, this retroreflecting geometry would result in two lattices accelerated in opposite directions.

During the interferometer phase, the retardance is set to zero, so that the Raman upward beams are linearly po- larized. This polarization arrangement allows to drive σ

₋

− σ

₋

transitions between the two |F = 1, m

F

= 0i and |F = 2, m

F

= 0i states. As a side effect, σ

₋

− σ

+

transitions are allowed, which couple |F = 1, m

F

= 0i to |F = 2, m

F

= −2i or |F = 2, m

F

= +2i depend- ing on the direction of the Raman effective wavevector.

This forces us to operate the interferometers with large bias fields of hundreds of mG, for which these parasitic transitions are non resonant.

With such a retroreflected geometry, the phase differ- ence between the Raman lasers is linked to the position of the mirror. Without isolation, fluctuations of this po- sition due to ground vibrations can induce significant in- terferometer phase noise, washing out the interferometer fringes, in the urban environment of the center of Paris.

This vibration noise is recorded with a low noise seis- mometer (Guralp 40T), placed next to the mirror, dur- ing the interferometer sequence. This allows to calculate an estimate of the common mode phase shift induced by parasitic vibrations onto the two interferometers. The correlation between the classical signal provided by the seismometer and the phase shifts of the quantum sensor can be exploited to post-correct the atomic measurement [6], to recover the interferometer fringes in the presence of a large noise [6, 10], or to correct the interferometer phase in real-time in order to keep the interferometer operating at mid-fringe where its sensitivity is maximal [11]. Given that our measurements are performed with- out any vibration isolation, the vibration noise quickly dominates over all sources of interferometer phase noise and can amount to several radians, even for the rela- tively short interferometer times we use here (2T is at most equal to 120 ms).

III. RESULTS

We start by illustrating the effect of the vibration noise onto the interferometer signal. Figure 4 displays the fringes recorded for two different interferometer times 2T = 2 ms and 2T = 70 ms. Here, the interferome- ter phase is scanned by varying the frequency chirp one needs to apply to the Raman laser frequency difference in order to compensate for the increasing Doppler effect and keep the Raman transitions resonant at each pulse.

While the fringes are clearly visible for 2T = 2 ms, they are washed out by the vibration noise for 2T = 70 ms.

The contrast of the fringes is significantly better for the

second cloud than for the first one, and gets lower for

larger T . This loss of contrast results mostly from the

expansion of the cloud and from its convolution with the

transverse intensity profile of the Raman lasers: coupling

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4

d)

b) c)

a)

0.4 0.5

0.20 0.25 0.30 0.35

Transitionprobabilityofcloud1

Transition probability of cloud 2

25.1441 25.1442 25.1443

0.2 0.3 0.4 0.5 0.6

Transitionprobability

Chirp rate (MHz/s) CLOUD 1

CLOUD 2

0.3 0.4 0.5 0.6 0.7

0.20 0.25 0.30 0.35 0.40

Transitionprobabilityofcloud1

Transition probability of cloud 2

24 25 26

0.2 0.3 0.4 0.5 0.6 0.7

Chirp rate (MHz/s) CLOUD 1

CLOUD 2

FIG. 4: Interferometer fringe patterns for the two simultaneous interferometers, for two different interferometer times 2 T = 2 ms (a) and 2 T = 70 ms (c). The interferometer phases are scanned by varying the frequency chirp applied to the Raman lasers.

The parametric plot of the transition probabilities for in phase (b) (resp. out of phase (d)) interferometers gives a line (resp.

an ellipse).

inhomogeneities get larger for larger atomic cloud sizes, and are thus larger for the launched cloud which has been expanding for much longer times compared to the sec- ond cloud. In the presence of large vibration noise, the phase fluctuations of the two interferometers are strongly correlated. This is evidenced by plotting the transition probabilities parametrically, which, in general and in par- ticular here for 2T = 70 ms, gives an ellipse (as illustrated in figure 4 (d)). Yet, when the two interferometers are in phase, the parametric plot gives a straight line as dis- played in figure 4 (b). A direct fit of the ellipse gives access to the differential phase, but the adjustment is in general biased, except when the differential phase is exactly π/2. In addition, such a fit cannot retrieve a null differential phase. In our experiment, the differen- tial phase between the two interferometers can easily be tuned by changing the amplitude of the bias magnetic field, as both interferometers have large, but different, phase shifts due to gradients of the applied magnetic field. Indeed, the magnetic field profiles are different across the atom trajectories of the two interferometers.

It varies for the first cloud from about 400 mG to 200 mG in between the first and last pulse of the interferometer, and for the second cloud oppositely from 200 mG to 400 mG.

Instead of the ellipse fitting method, we use here an- other method for the extraction of the differential phase.

Knowing the well calibrated scale factor of the seismome- ter and the transfer function of the interferometers versus acceleration noise [37], we calculate out of its signal an estimate of the common mode vibration phase shift ex- perienced by the two interferometers. We plot the mea- sured transition probabilities of the two interferometers as a function of this vibration induced phase shifts.

Figure 5 displays such plots, for given values of the bias magnetic fields and 2T = 120 ms. Here, visible fringe patterns are recovered, with the interferometer phase be- ing randomly scanned by the vibration noise. We can then fit each of these fringe pattern using the following formula:

P

i

= A

i

+ C

i

2 cos(D

i

× Φ

vib,S

+ Φ

i

) (1)

-8 -6 -4 -2 0 2 4 6 8

0.2 0.3 0.4 0.5 0.6 0.7

Cloud 1

Cloud 2

Seismic phase (rad)

FIG. 5: Measured transition probabilities versus vibration phase shift, estimated from the simultaneous measurement of a low noise seismometer. The total interferometer time is 2 T = 120 ms. The retrieval of the fringe pattern reveals the correlation between the interferometer phase fluctuations and the vibration noise recorded by the classical sensor.

with A

i

the offset, C

i

the contrast, Φ

vib,S

the calculated vibration induced phase shift and Φ

i

the phase shift of the i-th interferometer. D

i

is a coefficient which accounts for an eventual mismatch in the calibration of the seis- mometer. In practice, we find that D

1

and D

2

deviate from 1 by a few percents, due to the non flat response function of the seismometer [6]. Such fits allow to ex- tract the individual phases of the two interferometers, from which the differential phase is calculated.

To assess the sensitivity of this method, we divide a series of about 4000 consecutive measurements into pack- ets of 40 measurements, which we individually fit. The phases extracted from these fits are displayed on figure 6 for the two clouds. For these measurements, one can note that the fluctuations of the extracted phases, of or- der of about 0.6 rad peak-to-peak, are correlated for the two clouds.

We then calculate the Allan standard deviation (ASD)

of the fitted phase fluctuations of the two interferome-

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0 20 40 60 80 -0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Cloud 1

Cloud 2

Fittedphase(rad)

Packet number

FIG. 6: Interferometer phases extracted from consecutive fits of the fringes for subsets of 40 measurements. Total interfer- ometer time 2 T = 120 ms.

ters, and of their difference. Figure 7 displays such ASDs

1 10 40 100 1000

0.01 0.1 1

Allanstandarddeviation ofphasefluctuations(rad)

Number of measurements Cloud 1

Cloud 2

Difference

Seismic phase

FIG. 7: Allan standard deviations of the vibration phase noise (seismic phase displayed as open squares), the individual re- trieved interferometer phases (full squares and circles) and the differential phase (diamonds). Total interferometer time:

2 T = 120 ms.

for in phase interferometers with 2T = 120 ms. The ASDs average as white noise, and correspond to phase sensitivities of 150 mrad/packet of 40 shots, or equiva- lently of 1 rad/shot, for the individual interferometers.

In comparison, the ASD of the induced vibration noise of 2 rad/shot. The gain is not significant, due to the poor quality of the correlation, as can be seen in figure 5. Despite this, the ASD of the differential phase is sig- nificantly less, of about 33 mrad/packet (or equivalently of 208 mrad/shot), and lies not far from the limit set by our detection noise (120 mrad/shot on the differen- tial measurement). This puts into evidence the existence of strong correlations between the values of the fitted phases for the two interferometers, which are suppressed when taking their difference. From [32], we expect this

correlation, and thus the rejection of the common mode vibration noise, to decrease when the differential phase increases. We have investigated this loss of sensitivity by repeating this analysis for different differential phases in a range from -1.5 to 1.5 rad (this range corresponds to a variation of the current in the bias coils of 10%). Figure 8 displays the sensitivity of the differential phase extrac- tion we obtain as a function of the differential phase. In order to highlight the effect of this correlation, this sen- sitivity is normalized by the one we would expect in the absence of any correlation between the two interferom- eter phases, which correspond to the quadratic sum of the sensitivities σ

i

obtained individually: p

σ

1²

+ σ

2²

. We indeed observe that the sensitivity reaches its best level for in phase interferometers.

The results are compared with numerical simulations, which we take as representative as possible of the ex- periment. In these simulations, we generate the transi- tion probabilities of the two interferometers, by randomly drawing the vibration phase estimated by the seismome- ter Φ

vib,S

in a Gaussian distribution. We use for the con- trasts of the two interferometers the average values of the fitted contrasts, 10% and 6%. We add to the transition probabilities uncorrelated Gaussian detection noises with standard deviation σ

P

= 3 ×10

⁻³

, equal to the measured detection noise. We also randomly draw δΦ

vib

the differ- ence between the vibration phase Φ

vib

and its estimate Φ

vib,S

in a Gaussian distribution. The standard devia- tion of δΦ

vib

is adjusted so as to obtain the same sensitiv- ity of 1 rad/shot as in the measurements when extracting the individual phases from the simulated data using the method described above. This adjustment corresponds to a standard deviation of δΦ

vib

of 620 mrad/shot. We then repeat the simulations for various differential phases and the normalized sensitivity we obtain in the simulation for the extraction of the differential phase is displayed on figure 8 as a line. The shaded area corresponds to the uncertainty in the estimation of the Allan standard deviations, given that the number of data samples in the measurements is finite. This confidence area is estimated from the dispersion of the results obtained when repeat- ing numerical simulations with different sets of random draws and with the same number of data samples as for the measurements (typically 4000 measured samples for each differential phase, to be compared with the 500 000 draws generated to calculate the normalized sensitivity displayed as a line). We find a perfect agreement be- tween the experimental results and the corresponding simulation, given that most of our measurements lie in the shaded confidence area and that the uncertainties in the measured sensitivities match the width of the simu- lated confidence area.

We finally compare the technique studied here with the direct fit of the ellipse, using the same fitting procedure as [26], both in the measurements and the simulations.

Figure 9 displays the sensitivities of the differential phase

(not normalized here) obtained with these measurements,

displayed as points, and with the simulations, displayed

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6

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2

measurement

simulation

Normalizedsensitivity

Differential phase (rad)

FIG. 8: Sensitivity of the differential phase extraction as a function of the extracted differential phase. The points rep- resent the results of the measurements. The line corresponds to the result of the numerical simulation, and the shaded area to its uncertainty for a finite number of 4000 data samples.

as lines. For a quantitative match between experiments and simulation, we had for this simulation to decrease the amplitude of the vibration noise measured by the seis- mometer σ

Φ_vib,S

from 2 to 1.8 radian (the latter value corresponding to an average amplitude over all the mea- surements of the measured seismic noise, which fluctuates from one measurement to the other) and to increase the amplitude of σ

δΦ_vib

to 0.86 rad.

-3 -2 -1 0 1 2 3

0.00 0.05 0.10 0.15 0.20

Sensitivity(mrad/packet)

Extracted differential phase (rad)

FIG. 9: Comparison with the ellipse fitting technique. Points display the results of the measurements (squares: our method, circles: ellipse fitting), and lines of the simulation (black line:

our method, red (light gray) line: ellipse fitting).

With respect to the method presented here, the el- lipse fitting technique rejects better the common vibra- tion noise. Its sensitivity does not depend on the value of the differential phase, and equals the optimal sensi- tivity we obtain with our method for a null differential

phase. On the other hand, the simulations show that the ellipse fitting technique leads to a biased differen- tial phase, whereas our method is in principle unbiased.

Also, by contrast with our method, the ellipse fitting rou- tine cannot extract differential phases close to zero [26], and suffers from ambiguities in the determination of the differential phase, which complicates the extraction, es- pecially close to π/2. This explains the discontinuities of the red line in the figure 9. Finally, one can note signifi- cant deviations of the measurements from the simulation for the ellipse fitting. We attribute these mismatches to variations from one measurement to the other of the detection noise, due to changes in the contrast of the interferometers and in the number of detected atoms.

Assessing experimentally our ability of extracting the differential phase accurately, as claimed in [32], requires a method for varying this differential phase in an accurate way. This cannot easily be realised with magnetic field gradient phase shifts. An alternative method consists in changing the frequencies of the Raman lasers at the second pulse, such as demonstrated in [38].

IV. CONCLUSION

We have performed the experimental validation of the

method proposed in [32] for extracting the differential

phase in dual atom interferometers. The experiment was

performed on an atom gradiometer setup, consisting in

two simultaneous atom gravimeters separated along the

vertical direction. We have exploited the correlations be-

tween the individual noisy measurements of each interfer-

ometer and the estimates of the phase noise introduced

by parasitic ground vibrations to determine the individ-

ual phases of each interferometer, out of which the dif-

ferential phase is straight-forwardly obtained. We find

that the sensitivity of the differential phase extraction is

optimal, and close to the limit set by the detection noise,

when the two interferometers are in phase. We have fi-

nally briefly compared this method with the simple and

more often used ellipse-fitting method. A thorough com-

parison with other techniques would be of interest, but

lie beyond the scope of this paper. In the future, we will

demonstrate the accuracy of this method, thanks to the

fine tuning of the differential phase obtained by chang-

ing the frequency of the Raman lasers at the central π

pulse. As shown in [39] and pointed out in [40], with a

specific adjustement of this frequency change and thus

of the corresponding Raman wavevector, the gradiome-

ter differential phase can be compensated, which allows

for a precise determination of the gravity gradient inde-

pendently of the gradiometric baseline. Our method for

extracting the differential phase thus appears perfectly

suited to the implementation of this compensation tech-

nique since it works best for a null differential phase.

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V. ACKNOWLEDGEMENTS

This work was supported by CNES (R&T R-S15/SU- 0001-048), by DGA (Gradiom project), by the “Domaine d’Intérêt Majeur” NanoK of the Région Ile-de-France, by the CNRS program “Gravitation, Références, As- tronomie, Métrologie” (PN-GRAM) co-funded by CNES

and by the LABEX Cluster of Excellence FIRST-TF (ANR-10-LABX-48-01), within the Program “Investisse- ments d’Avenir” operated by the French National Re- search Agency (ANR). M.L. thanks Muquans for finan- cial support.

M.L. and R.C. contributed equally to this work.

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